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Experimental Design — Definition, Formula & Examples

Experimental design is the plan for running an experiment. It decides who gets which treatment and what gets measured, so you can tell whether a treatment actually causes a change.

An experimental design specifies the experimental units, the explanatory variables (factors) and their levels, the treatments (combinations of factor levels), the chance mechanism that assigns treatments to units, and the response variable to be measured. A well-designed experiment follows four principles: comparison of two or more treatments, random assignment of treatments to units, control of other variables that could affect the response, and replication on enough units to estimate chance variation.

Key Formula

number of treatments=L1×L2×⋯×Lk\text{number of treatments} = L_1 \times L_2 \times \cdots \times L_k
Where:
  • kk = number of factors in the experiment
  • LiL_i = number of levels of factor i

How It Works

Start by naming the factors, their levels, and the response variable. Multiply the numbers of levels to count the treatments. Then use a chance process to assign units to treatments, such as numbering the units and using a random number generator. In a completely randomized design, all units are assigned at random. In a randomized block design, you first group similar units into blocks, then randomize within each block. A matched pairs design is a special block design with blocks of size two. Because randomization tends to balance lurking variables across groups, a difference in responses that is too large to be due to chance alone can be attributed to the treatments.

Worked Example

Problem: A researcher has 60 tomato plants. She wants to test 3 fertilizer types (A, B, C) and 2 watering schedules (daily, weekly) on plant height after 8 weeks. Design a completely randomized experiment with equal group sizes.
Step 1: Identify the parts. The experimental units are the 60 plants. The factors are fertilizer (3 levels) and watering (2 levels). The response variable is height after 8 weeks.
Step 2: Count the treatments. Each treatment is one fertilizer paired with one watering schedule.
3×2=6 treatments3 \times 2 = 6 \text{ treatments}
Step 3: Find the group size so that each treatment is replicated equally.
606=10 plants per treatment\frac{60}{6} = 10 \text{ plants per treatment}
Step 4: Randomize. Number the plants 1 to 60. Use a random number generator to pick 10 distinct numbers for treatment 1 (A, daily). Pick 10 more from those left for treatment 2, and continue. The last 10 plants get treatment 6. Keep sunlight, pot size, and soil the same for every plant. After 8 weeks, compare the mean heights of the six groups.
Answer: The experiment has 6 treatments, with 10 randomly assigned plants each. You then compare mean plant height across the six groups.

Why It Matters

AP Statistics free-response questions often ask you to describe a randomization method or to explain why a block design fits a situation. The same principles govern clinical drug trials run under FDA rules, A/B tests at tech companies, and crop-yield studies in agriculture. Without a sound design, no p-value or confidence interval can justify a cause-and-effect claim.

Common Mistakes

Mistake: Confusing random assignment with random sampling.
Correction: Random sampling picks who is in the study and lets you generalize to a population. Random assignment decides who gets which treatment and is what allows cause-and-effect conclusions. An experiment can use volunteers (no random sample) and still support causal claims about those subjects.
Mistake: Writing 'randomly assign the subjects' with no method described.
Correction: Describe a specific chance process. For example: number subjects 1 to 60, use a random number generator to choose 30 distinct numbers for treatment 1, and give the rest treatment 2. Drawing names from a hat after mixing the slips well also works.

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